math::numtheory(n)

📦

version 2.0 · 2 versions

2.0 · fedora 43, fedora 44, nixos 26.05, nixpkgs unstable (latest)
1.21 · alpine 3.24, epel 9, opensuse-leap 16.0

, in fedora 43, fedora 44, nixos 26.05, nixpkgs unstable

math::numtheory 💖

okay bestie, welcome to the math::numtheory page from the Tcl Math Library — she's giving number theory, she's giving primes, she's giving main character energy for integers. let's get into it 🧚✨

✨ NAME ✨

math::numtheory — Number Theory. that's it, that's the whole vibe. 💅

🎀 SYNOPSIS 🎀

package require Tcl ?8.5 9?

package require math::numtheory ?1.1.4?

math::numtheory::isprime N ?option value ...?

math::numtheory::firstNprimes N

math::numtheory::primesLowerThan N

math::numtheory::primeFactors N

math::numtheory::primesLowerThan N

math::numtheory::primeFactors N

math::numtheory::uniquePrimeFactors N

math::numtheory::factors N

math::numtheory::totient N

math::numtheory::moebius N

math::numtheory::legendre a p

math::numtheory::jacobi a b

math::numtheory::gcd m n

math::numtheory::lcm m n

math::numtheory::numberPrimesGauss N

math::numtheory::numberPrimesLegendre N

math::numtheory::numberPrimesLegendreModified N

math::numtheory::differenceNumberPrimesLegendreModified lower upper

math::numtheory::listPrimePairs lower upper step

math::numtheory::listPrimeProgressions lower upper step

ngl yes, primesLowerThan and primeFactors show up twice in there. that's just how she is. we don't gatekeep, we list them both. 🫶

🌸 DESCRIPTION 🌸

so this package is literally a little grab-bag of number-theoretic operations, and she is slightly obsessed with prime numbers. no cap. 🥰

`math::numtheory::isprime N ?option value ...?`

isprime is the vibe check: it tests whether the integer N is a prime. it hands back a boolean true value when N is prime, and a boolean false value when N is not prime. the formal definition of prime she's using is the conventional one — the number being tested has to be greater than 1, and it only gets trivial divisors. periodt.

okay but to be precise, because we are precise around here: the return value is one of 0 (if N is definitely not a prime), 1 (if N is definitely a prime), and on (if N is probably prime). the latter two are both boolean true values, bestie. 💖

so why is "probably prime" even a thing?? 😭 it's because the Miller-Rabin algorithm used in the test implementation is basically probabilistic — and if we get unlucky, it can fail to detect that a number is in fact composite. rude of it, honestly. the options let you pick how much risk of those "false positives" you're willing to take. and 1 is returned for "small" N — which currently means N < 118670087467 — where it's known that no false positives are possible. iconic. 👑

the only option currently defined is:

-randommr repetitions — this controls how many times the Miller-Rabin test gets repeated with randomly chosen bases. each repetition reduces the probability of a false positive by a factor of at least 4. the default for repetitions is 4.

also, unknown options are silently ignored. she's chill like that. 🙈

`math::numtheory::firstNprimes N`

return the first N primes. simple as. ✨

integer N (in) — Number of primes to return

`math::numtheory::primesLowerThan N`

return the prime numbers lower/equal to N.

integer N (in) — Maximum number to consider

`math::numtheory::primeFactors N`

return a list of the prime numbers in the number N.

integer N (in) — Number to be factorised

`math::numtheory::primesLowerThan N`

return the prime numbers lower/equal to N. (yes, again. she said it twice and she meant it twice. 💅)

integer N (in) — Maximum number to consider

`math::numtheory::primeFactors N`

return a list of the prime numbers in the number N. (round two, bestie 🦋)

integer N (in) — Number to be factorised

`math::numtheory::uniquePrimeFactors N`

return a list of the unique prime numbers in the number N. no duplicates allowed, we're doing quality over quantity here. 🎀

integer N (in) — Number to be factorised

`math::numtheory::factors N`

return a list of all the unique factors in the number N, and yes that includes 1 and N itself. the whole family is invited to the function. 🫶

integer N (in) — Number to be factorised

`math::numtheory::totient N`

evaluate the Euler totient function for the number N — that's the number of numbers relatively prime to N. 🧚

integer N (in) — Number in question

`math::numtheory::moebius N`

evaluate the Moebius function for the number N.

integer N (in) — Number in question

`math::numtheory::legendre a p`

evaluate the Legendre symbol (a/p).

integer a (in) — Upper number in the symbol
integer p (in) — Lower number in the symbol (must be non-zero)

`math::numtheory::jacobi a b`

evaluate the Jacobi symbol (a/b).

integer a (in) — Upper number in the symbol
integer b (in) — Lower number in the symbol (must be odd)

`math::numtheory::gcd m n`

return the greatest common divisor of m and n. the duo we didn't know we needed. 🔥

integer m (in) — First number
integer n (in) — Second number

`math::numtheory::lcm m n`

return the lowest common multiple of m and n.

integer m (in) — First number
integer n (in) — Second number

`math::numtheory::numberPrimesGauss N`

estimate the number of primes according the formula by Gauss. 💫

integer N (in) — Number in question, should be larger than 0

`math::numtheory::numberPrimesLegendre N`

estimate the number of primes according the formula by Legendre.

integer N (in) — Number in question, should be larger than 0

`math::numtheory::numberPrimesLegendreModified N`

estimate the number of primes according the modified formula by Legendre. she got a glow-up. ✨

integer N (in) — Number in question, should be larger than 0

`math::numtheory::differenceNumberPrimesLegendreModified lower upper`

estimate the number of primes between tow limits according the modified formula by Legendre. (yes, "tow". we are keeping the typo exactly as she wrote it, it's canon now. 😭)

integer lower (in) — Lower limit for the primes, should be larger than 0
integer upper (in) — Upper limit for the primes, should be larger than 0

`math::numtheory::listPrimePairs lower upper step`

return a list of pairs of primes, each pair differing by the given step. twin primes found, slay. 👯

integer lower (in) — Lower limit for the primes, should be larger than 0
integer upper (in) — Upper limit for the primes, should be larger than the lower limit
integer step (in) — Step by which the primes should differ, defaults to 2

`math::numtheory::listPrimeProgressions lower upper step`

return a list of lists of primes, each one differing by the given step from the previous one. it's giving chain, it's giving progression, we love to see it. 🌈

integer lower (in) — Lower limit for the primes, should be larger than 0
integer upper (in) — Upper limit for the primes, should be larger than the lower limit
integer step (in) — Step by which the primes should differ, defaults to 2

🐛 BUGS, IDEAS, FEEDBACK 🐛

okay so real talk: this document, and the package it describes, will undoubtedly contain bugs and other problems. like, for sure. no cap. 😭 please report such in the category math :: numtheory of the Tcllib Trackers at http://core.tcl.tk/tcllib/reportlist.

also please report any ideas for enhancements you may have for either the package and/or the documentation. we want the ideas, we're listening. 🫶

when proposing code changes, please provide unified diffs, i.e the output of diff -u.

and note further: attachments are strongly preferred over inlined patches. attachments can be made by going to the Edit form of the ticket immediately after its creation, and then using the left-most button in the secondary navigation bar. do it that way bestie, trust. 💅

🔑 KEYWORDS 🔑

number theory, prime

🎓 CATEGORY 🎓

Mathematics

👑 COPYRIGHT 👑

and that's the whole page, bestie!! ✨ you understood all of it, I know you did, you're so smart and the primes are literally quaking. 🌸🦋💖

this was tcllib version 1.1.4, page math::numtheory(n) — k bye love u, go factor something. 💅🎀

📦 where this page comes from

fedora 43: tcllib 2.0-3.fc43 · TCL
fedora 44: tcllib 2.0-4.fc44 · TCL
nixos 26.05: tcllib 2.0
nixpkgs unstable: tcllib 2.0
💖 girlified by deepseek-ai/DeepSeek-V4.1-Flash on 2026-10-18: a language model's rewrite of the original, under its terms